# Fibonacci proof by induction

1. Dec 7, 2008

### kathrynag

1. The problem statement, all variables and given/known data
$$F_{1}+F_{3}+F_{2n-1}$$=$$F_{2n}$$

2. Relevant equations

3. The attempt at a solution
P(k+1):$$F_{2k-1}+F_{2k+1}$$=$$F_{2k+2}$$

2. Dec 7, 2008

### mutton

What happened to F_1 and F_3?

3. Dec 7, 2008

### kathrynag

well, it's F_1+F_3+....+F_2n-1

4. Dec 7, 2008

### mutton

Then P(k + 1) needs to be changed accordingly.

What have you tried so far?

5. Dec 7, 2008

### kathrynag

I've tried plugging in numbers.

6. Dec 7, 2008

### mutton

And how did that work out?

The definition of Fibonacci numbers will be helpful in the induction proof.

7. Dec 8, 2008

### kathrynag

If I plug in 1, I just get F_1, so 1=1
If I plug in 2, I get F_3, so 1+2=F_4, 3=3

8. Dec 8, 2008

### kathrynag

F-1+F-3+...+F_2k-1+F_2k+1
P(k)+F_2k+1
F_2k+F_2k+1

Now I'm stumped...

9. Dec 8, 2008

### HallsofIvy

Staff Emeritus
What was "F_1+ F_3+ ...+ F_2n-1"?

In your first post you said the problem was to prove that
$$F_{1}+F_{3}+F_{2n-1}$$=$$F_{2n}$$

Are you saying now it is actually to prove that
$$F_{1}+F_{3}+\cdot\cdot\cdot +F_{2n-1}$$=$$F_{2n}$$?

10. Dec 8, 2008

### mutton

Very close. What happens when 2 consecutive Fibonacci numbers are added?

11. Dec 8, 2008

### kathrynag

It equals the 3rd Fibonnacci number.
F_1+F-2=F_3

so F_2k+F_2k+1=F_2k+1+1

12. Dec 8, 2008

### mutton

And that's exactly what you wanted to show.

13. Dec 8, 2008

### kathrynag

Ok thanks!

14. Dec 9, 2008

### epenguin

Sorry but maybe the problem is not properly stated? Are the F defined to be Fibonacci numbers?

Then F2n = F2n-1 + F2n-2

So if you are then asking also that

F2n = F2n-1 + F1 + F3

then

F2n-2 = F1 + F3

which is not making much sense.